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4B · Physical principles of living systems
Fluids, circulation, and gas exchange
Fluids at rest (pressure, Pascal's principle, buoyancy) and in motion (the continuity equation, Bernoulli's principle, and Poiseuille resistance), applied to the body's plumbing — the circulatory system and gas exchange in the lungs.
Fluid statics
A static fluid exerts pressure that increases with depth (P = ρgh), transmits applied pressure equally in all directions (Pascal), and buoys submerged objects by the weight of fluid they displace (Archimedes).
Pressure is force per area (P = F/A, in pascals). In a fluid column the gauge pressure at depth h is ρgh, and the absolute pressure adds the pressure pushing down on the surface (P = P₀ + ρgh). Pressure at a given depth is the same in every direction and depends only on depth, not on container shape.
Density and specific gravity
Density ρ = m/V (water ≈ 1000 kg/m³ = 1 g/cm³). Specific gravity is density relative to water — so an object with SG < 1 floats in water, SG > 1 sinks.
Specific gravity is dimensionless and makes buoyancy problems fast: the fraction of a floating object that sits submerged equals its specific gravity (ice at SG ≈ 0.92 floats with ~92% underwater). Density also sets the hydrostatic pressure (ρgh) and the buoyant force, so it threads through the whole chapter.
Hydrostatic pressure
Gauge pressure at depth h is P = ρgh — it depends only on depth and fluid density, not on the shape or width of the container.
Because pressure depends only on depth, two connected columns of different widths reach the same pressure at the same height (the hydrostatic paradox). This is why blood pressure readings are referenced to heart level, and why standing up drops the pressure of blood reaching the brain. Atmospheric pressure (~101 kPa ≈ 760 mmHg) is the P₀ term added to get absolute pressure.
Pascal's principle
Pressure applied to an enclosed fluid is transmitted undiminished to every point — the basis of the hydraulic lift, which multiplies force at the cost of distance.
In a hydraulic system P is constant, so F₁/A₁ = F₂/A₂: a small force on a small piston produces a large force on a large piston (force multiplied by the area ratio). As with levers, the trade-off is distance — the large piston moves proportionally less, conserving work.
Related
Same force-for-distance bargain as the body's levers.
Buoyancy and Archimedes' principle
The buoyant force equals the weight of the displaced fluid: F_b = ρ_fluid · V_displaced · g. It depends on the fluid's density and the submerged volume — not on the object's weight.
An object floats when it can displace its own weight in fluid before fully submerging (which requires ρ_object < ρ_fluid); it sinks otherwise. For a floating object, V_submerged/V_total = ρ_object/ρ_fluid. For a fully submerged object, the buoyant force is fixed (the whole volume is displaced) and the object rises, sinks, or hovers depending on whether ρ_object is less than, greater than, or equal to ρ_fluid.
Don't confuse
Buoyant force depends on the fluid's density and the displaced volume, not on the object's mass or what it's made of. Two objects of equal volume fully submerged feel the same buoyant force even if one is lead and one is wood — what differs is their weight, hence whether they sink.
Fluid dynamics
For moving fluids: the continuity equation (A·v is constant) says a fluid speeds up where a tube narrows; Bernoulli's principle says faster flow means lower pressure; and Poiseuille's law says flow through a tube scales with the fourth power of the radius.
Idealized flow is incompressible and non-viscous; real flow (like blood) has viscosity and resistance. The two ideal laws — continuity and Bernoulli — handle the geometry and energy of flow, while Poiseuille's law adds the resistance that real, viscous flow encounters in a tube.
The continuity equation
Conservation of mass for an incompressible fluid: A₁v₁ = A₂v₂. Flow rate (Q = A·v) is constant, so where the cross-section narrows, the fluid speeds up.
The volume flow rate Q (m³/s) is the same at every cross-section of a closed system. Halve the radius and area drops to a quarter, so velocity must rise four-fold to keep Q constant. This is why a partially blocked (stenosed) artery has faster flow through the narrowing, and why a thumb over a hose nozzle speeds up the stream.
How AAMC tests it
A vessel narrows to half its diameter — what happens to flow speed? Area ∝ diameter², so a half-diameter is a quarter-area, and speed quadruples.
Bernoulli's principle
For ideal flow, energy per volume is conserved: P + ½ρv² + ρgh = constant. At the same height, faster-moving fluid has lower pressure.
Bernoulli's equation is conservation of energy for a fluid: a pressure term, a kinetic term (½ρv²), and a gravitational term (ρgh). Hold height constant and the kinetic and pressure terms trade off — so in a constriction, where continuity forces the fluid faster, the pressure drops. This underlies lift on a wing, the lateral pull in an aneurysm or stenosis, and the operation of a venturi or atomizer.
Don't confuse
Faster flow = lower pressure (Bernoulli), which feels backward — a narrowing speeds the fluid and lowers its pressure. Don't reason that "squeezing" the fluid raises its pressure; continuity raises its speed, and Bernoulli lowers its pressure.
Viscosity and Poiseuille's law
For viscous flow through a tube, Q = ΔP·πr⁴ / (8ηL) — flow scales with the fourth power of the radius. Resistance R = 8ηL/(πr⁴), so halving the radius cuts flow 16-fold.
Real fluids resist flow through viscosity (η). Poiseuille's law shows that of all the variables, radius dominates because of the r⁴ term: a small vasoconstriction produces a large drop in flow (and a large rise in the pressure needed to maintain it). This is the lever the body uses to regulate blood distribution and blood pressure, and why arteriolar tone matters so much. Flow also rises with the pressure gradient ΔP and falls with tube length and viscosity.
Don't confuse
Flow depends on radius to the fourth power, not linearly or squared. The instinct to treat a "half-width tube" as "half the flow" is the classic miss — it's 1/16 the flow.
Related
Contrast the ideal, frictionless picture in Bernoulli's principle: Poiseuille is what's left once viscosity is included.
Laminar vs. turbulent flow
Laminar flow is smooth, ordered layers (and is what Poiseuille describes); turbulent flow is chaotic and sets in past a critical velocity (high Reynolds number). Turbulence is audible — it's the source of bruits and Korotkoff sounds.
Laminar flow streams in parallel layers, fastest at the center and zero at the walls. Above a critical velocity — predicted by a high Reynolds number (favored by high velocity, large radius, high density, low viscosity) — flow becomes turbulent, raising resistance and generating sound. Blood is normally laminar (silent); turbulence at a narrowed valve or a blood-pressure cuff is what the stethoscope hears.
Circulation and gas exchange
The circulatory and respiratory systems are fluid-mechanics problems: blood flow obeys ΔP = Q·R (the circulatory analog of Ohm's law), and gas exchange is diffusion down partial-pressure gradients across the alveolar and capillary membranes.
Treat the circulation as a pressure-driven flow network: cardiac output (Q) times total peripheral resistance (R) sets the arterial pressure (ΔP = QR), and the arterioles — via the r⁴ term — are the main resistance valves. In the lungs, gas exchange runs on partial pressures: O₂ diffuses from high alveolar partial pressure into blood, CO₂ the other way, each moving down its own gradient. The amount of gas dissolving is set by its partial pressure (Henry's law), a point of overlap with the solutions chemistry in FC5.
Blood pressure and vascular resistance
ΔP = Q·R (the hemodynamic Ohm's law). Because resistance scales with 1/r⁴, small changes in arteriolar radius (vasoconstriction/dilation) produce large changes in blood pressure and flow distribution.
Mean arterial pressure equals cardiac output times total peripheral resistance. The arterioles are the principal site of resistance and regulation: a modest vasoconstriction sharply raises resistance (r⁴) and redirects flow. Pressure also falls along the circuit — highest in the aorta, lowest in the veins — and the large total cross-sectional area of the capillaries slows flow there (continuity again), maximizing time for exchange.
Related
The resistance term is Poiseuille's 0 ; the slow capillary flow is continuity over a large area.
Gas exchange and partial pressures
Gases diffuse down partial-pressure gradients: O₂ from alveoli into blood, CO₂ from blood into alveoli. The dissolved amount of a gas is proportional to its partial pressure (Henry's law).
Each gas in a mixture exerts a partial pressure (Dalton's law: partial pressures sum to the total). In the lung, alveolar O₂ partial pressure exceeds that of venous blood, so O₂ diffuses in; CO₂'s gradient runs the other way. Diffusion rate rises with surface area and the partial-pressure gradient and falls with membrane thickness (Fick's law) — which is why emphysema (less area) and fibrosis (thicker membrane) impair exchange. Henry's law (dissolved gas ∝ partial pressure) connects this to the solution chemistry of FC5.
Worked question
A segment of artery narrows so that its radius is reduced to one-half of its original value. Assuming the pressure gradient across the segment is unchanged, by what factor does the volume flow rate through that segment change?